Optimal. Leaf size=37 \[ \frac {2 \cosh (x)}{243 \sqrt {4 \cosh ^2(x)-9}}-\frac {\cosh (x)}{27 \left (4 \cosh ^2(x)-9\right )^{3/2}} \]
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Rubi [A] time = 0.05, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3190, 192, 191} \[ \frac {2 \cosh (x)}{243 \sqrt {4 \cosh ^2(x)-9}}-\frac {\cosh (x)}{27 \left (4 \cosh ^2(x)-9\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 192
Rule 3190
Rubi steps
\begin {align*} \int \frac {\sinh (x)}{\left (-9+4 \cosh ^2(x)\right )^{5/2}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{\left (-9+4 x^2\right )^{5/2}} \, dx,x,\cosh (x)\right )\\ &=-\frac {\cosh (x)}{27 \left (-9+4 \cosh ^2(x)\right )^{3/2}}-\frac {2}{27} \operatorname {Subst}\left (\int \frac {1}{\left (-9+4 x^2\right )^{3/2}} \, dx,x,\cosh (x)\right )\\ &=-\frac {\cosh (x)}{27 \left (-9+4 \cosh ^2(x)\right )^{3/2}}+\frac {2 \cosh (x)}{243 \sqrt {-9+4 \cosh ^2(x)}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 26, normalized size = 0.70 \[ \frac {\cosh (x) (4 \cosh (2 x)-23)}{243 (2 \cosh (2 x)-7)^{3/2}} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sinh (x)}{\left (-9+4 \cosh ^2(x)\right )^{5/2}} \, dx \]
Verification is Not applicable to the result.
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fricas [B] time = 0.94, size = 474, normalized size = 12.81 \[ \frac {2 \, \cosh \relax (x)^{8} + 16 \, \cosh \relax (x) \sinh \relax (x)^{7} + 2 \, \sinh \relax (x)^{8} + 28 \, {\left (2 \, \cosh \relax (x)^{2} - 1\right )} \sinh \relax (x)^{6} - 28 \, \cosh \relax (x)^{6} + 56 \, {\left (2 \, \cosh \relax (x)^{3} - 3 \, \cosh \relax (x)\right )} \sinh \relax (x)^{5} + 2 \, {\left (70 \, \cosh \relax (x)^{4} - 210 \, \cosh \relax (x)^{2} + 51\right )} \sinh \relax (x)^{4} + 102 \, \cosh \relax (x)^{4} + 8 \, {\left (14 \, \cosh \relax (x)^{5} - 70 \, \cosh \relax (x)^{3} + 51 \, \cosh \relax (x)\right )} \sinh \relax (x)^{3} + 4 \, {\left (14 \, \cosh \relax (x)^{6} - 105 \, \cosh \relax (x)^{4} + 153 \, \cosh \relax (x)^{2} - 7\right )} \sinh \relax (x)^{2} - 28 \, \cosh \relax (x)^{2} + 8 \, {\left (2 \, \cosh \relax (x)^{7} - 21 \, \cosh \relax (x)^{5} + 51 \, \cosh \relax (x)^{3} - 7 \, \cosh \relax (x)\right )} \sinh \relax (x) + {\left (2 \, \cosh \relax (x)^{6} + 12 \, \cosh \relax (x) \sinh \relax (x)^{5} + 2 \, \sinh \relax (x)^{6} + 3 \, {\left (10 \, \cosh \relax (x)^{2} - 7\right )} \sinh \relax (x)^{4} - 21 \, \cosh \relax (x)^{4} + 4 \, {\left (10 \, \cosh \relax (x)^{3} - 21 \, \cosh \relax (x)\right )} \sinh \relax (x)^{3} + 3 \, {\left (10 \, \cosh \relax (x)^{4} - 42 \, \cosh \relax (x)^{2} - 7\right )} \sinh \relax (x)^{2} - 21 \, \cosh \relax (x)^{2} + 6 \, {\left (2 \, \cosh \relax (x)^{5} - 14 \, \cosh \relax (x)^{3} - 7 \, \cosh \relax (x)\right )} \sinh \relax (x) + 2\right )} \sqrt {\frac {2 \, \cosh \relax (x)^{2} + 2 \, \sinh \relax (x)^{2} - 7}{\cosh \relax (x)^{2} - 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}}} + 2}{486 \, {\left (\cosh \relax (x)^{8} + 8 \, \cosh \relax (x) \sinh \relax (x)^{7} + \sinh \relax (x)^{8} + 14 \, {\left (2 \, \cosh \relax (x)^{2} - 1\right )} \sinh \relax (x)^{6} - 14 \, \cosh \relax (x)^{6} + 28 \, {\left (2 \, \cosh \relax (x)^{3} - 3 \, \cosh \relax (x)\right )} \sinh \relax (x)^{5} + {\left (70 \, \cosh \relax (x)^{4} - 210 \, \cosh \relax (x)^{2} + 51\right )} \sinh \relax (x)^{4} + 51 \, \cosh \relax (x)^{4} + 4 \, {\left (14 \, \cosh \relax (x)^{5} - 70 \, \cosh \relax (x)^{3} + 51 \, \cosh \relax (x)\right )} \sinh \relax (x)^{3} + 2 \, {\left (14 \, \cosh \relax (x)^{6} - 105 \, \cosh \relax (x)^{4} + 153 \, \cosh \relax (x)^{2} - 7\right )} \sinh \relax (x)^{2} - 14 \, \cosh \relax (x)^{2} + 4 \, {\left (2 \, \cosh \relax (x)^{7} - 21 \, \cosh \relax (x)^{5} + 51 \, \cosh \relax (x)^{3} - 7 \, \cosh \relax (x)\right )} \sinh \relax (x) + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.73, size = 40, normalized size = 1.08 \[ -\frac {{\left ({\left (2 \, e^{\left (2 \, x\right )} - 21\right )} e^{\left (2 \, x\right )} - 21\right )} e^{\left (2 \, x\right )} + 2}{486 \, {\left (e^{\left (4 \, x\right )} - 7 \, e^{\left (2 \, x\right )} + 1\right )}^{\frac {3}{2}}} + \frac {1}{243} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 30, normalized size = 0.81
method | result | size |
derivativedivides | \(-\frac {\cosh \relax (x )}{27 \left (-9+4 \left (\cosh ^{2}\relax (x )\right )\right )^{\frac {3}{2}}}+\frac {2 \cosh \relax (x )}{243 \sqrt {-9+4 \left (\cosh ^{2}\relax (x )\right )}}\) | \(30\) |
default | \(-\frac {\cosh \relax (x )}{27 \left (-9+4 \left (\cosh ^{2}\relax (x )\right )\right )^{\frac {3}{2}}}+\frac {2 \cosh \relax (x )}{243 \sqrt {-9+4 \left (\cosh ^{2}\relax (x )\right )}}\) | \(30\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.58, size = 125, normalized size = 3.38 \[ -\frac {1855 \, e^{\left (-2 \, x\right )} - 8485 \, e^{\left (-4 \, x\right )} + 5285 \, e^{\left (-6 \, x\right )} - 980 \, e^{\left (-8 \, x\right )} + 56 \, e^{\left (-10 \, x\right )} - 106}{12150 \, {\left (3 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} + 1\right )}^{\frac {5}{2}} {\left (-3 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} + 1\right )}^{\frac {5}{2}}} + \frac {980 \, e^{\left (-2 \, x\right )} - 5285 \, e^{\left (-4 \, x\right )} + 8485 \, e^{\left (-6 \, x\right )} - 1855 \, e^{\left (-8 \, x\right )} + 106 \, e^{\left (-10 \, x\right )} - 56}{12150 \, {\left (3 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} + 1\right )}^{\frac {5}{2}} {\left (-3 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} + 1\right )}^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 57, normalized size = 1.54 \[ -\frac {{\mathrm {e}}^x\,\sqrt {4\,{\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}^2-9}\,\left (21\,{\mathrm {e}}^{2\,x}+21\,{\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{6\,x}-2\right )}{486\,{\left ({\mathrm {e}}^{4\,x}-7\,{\mathrm {e}}^{2\,x}+1\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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